Does Mathematica provide a way to take, for instance:

ImplicitRegion[x^2 + y^2 <= 1, {x,y}]

and convert it into a region defined explicitly in terms of x and y:

-1 <= x <= 1 && -Sqrt[1-x^2] <= y <= Sqrt[1-x^2]


I found Resolve but that only seems to be able to project onto a plane or line.

  • 2
    $\begingroup$ Does Reduce@RegionMember[ImplicitRegion[x^2 + y^2 <= 1, {x, y}], {x, y}] meet your needs? $\endgroup$ – Simon Woods Nov 22 '16 at 22:57
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    $\begingroup$ See CylindricalDecomposition . The example in the doc is exactly what you ask (!) $\endgroup$ – andre314 Nov 22 '16 at 23:09
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    $\begingroup$ I suspect the applications are far more general than the specific disk illustrated in the question, and thus CylindricalDecomposition will have very limited use. $\endgroup$ – David G. Stork Nov 22 '16 at 23:16
  • $\begingroup$ @andre Answer? Even the usage message of CylindricalDecomposition seems to describe exactly the OP's problem. "convert it into a region defined explicitly in terms of x and y:", "a decomposition of the region represented by the inequalities ineqs into cylindrical parts whose directions correspond to the successive Subscript[x, i]." $\endgroup$ – Szabolcs Nov 23 '16 at 11:36
  • $\begingroup$ If someone wants to study a eventual CylindricalDecomposition solution, and give an answer, there's no problem. I don't have time at the moment. $\endgroup$ – andre314 Nov 23 '16 at 12:50

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